Set Theory

A translation of the π-calculus into the MONSTR graph rewriting language is described and proved correct. The translation illustrates the heavy cost in practice of faithfully implementing the communication primitive of the π-calculus and similar process calculi. It also illustrates the convenience of representing an evolving network of communicating agents directly within a graph manipulation formalism, both because the necessity to use delicate notions of bound variables and of scopes is avoided, and also because the standard model of graphs in set theory automatically yields a useful semantics for the process calculus. The correctness proof illustrates many features typically encountered in reasoning about graph rewriting systems, and particularly how serialisation techniques can be used to reorder an arbitrary execution into one having stated desirable properties.

Traditional agenda-setting theory is about the influence of mass media on the public's focus of attention, who and what people are thinking about. The expanded theory of agenda setting tested here during the 1995 regional and municipal elections in Spain ...

Is attention both necessary and sufficient for consciousness? Call this central question of this treatise, “Q.” We commonly have the experience of consciously paying attention to something, but is it possible to be conscious of something you are not attending to, or to attend to something of which you are not conscious? Where might we find examples of these?
This treatise is a quest to find an answer to Q in two parts. Part I reviews the foundations upon which the discourse on Q is built. Different inputs to Q produce different answers. After consideration of the many ways “attention” and “consciousness” have been defined, I settle upon phenomenal consciousness and Executive Attention (defined as a suite of strategies for structuring cognition for further processing implemented by the executive of working memory) as the most interesting inputs to Q, and the ones on which Part II focuses.
Attention without consciousness seems relatively easy to establish empirically, but consciousness without attention is much harder. The putative candidates all seem to have major problems, but I build a strong abductive case for the hitherto ignored case of foveal phenomenal overflow. We consciously see far more detail in our foveal fields than we can Executively Attend, although there is a serious obstacle to our ever confirming that empirically—identifying conscious content relies on Executive Attentional report. Triangulating the capacity limitations of attention, consciousness, and working memory strengthens this case for consciousness without attention, and suggests that cognition must work something like my “Witches’ Hat Model,” on which content can become conscious outside of Executive Attention or working memory. I conclude with some reflections on the implications of my arguments for the discourse on Q, and for other discourses such as the ontologies of attention and consciousness, theories of consciousness, some other cognitive concepts, and ethical considerations in humans, animals, and machines. A conclusive answer to Q continues to elude us. It may perhaps be an ultimately insoluble conundrum. But it is the very essence of humanity to seek an answer, and in so doing, to improve our understanding of our own nature:
“The proper study of mankind is man.”

The Continuum Problem has inspired set theorists and philosophers since the days of Cantorian set theory. In the last 15 years, W. Hugh Woodin, a leading set theorist, has not only taken it upon himself to engage in this question, he has also changed his mind about the answer. This paper illustrates Woodin’s solutions to the problem, starting in Sect. 3 with his 1999–2004 argument that Cantor’s hypothesis about the continuum was incorrect. From 2010 onwards, Woodin presents a very different argument, an argument that Cantor’s hypothesis is in fact true. This argument is still incomplete, but according to Woodin, some of the philosophical issues surrounding the Continuum Problem have been reduced to precise mathematical questions, questions that are, unlike Cantor’s hypothesis, solvable from our current theory of sets.

A \emph{standard formalization} of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $\in$). Patrick Suppes (\cite{sup92}) expressed skepticism about whether there is a ``simple or elegant method'' for presenting mathematicized scientific theories in such a standard formalization, because they ``assume a great deal of mathematics as part of their substructure''.

The major difficulties amount to these. First, as the theories of interest are \emph{mathematicized}, one must specify the underlying \emph{applied mathematics base theory}, which the physical axioms live on top of. Second, such theories are typically \emph{geometric}, concerning quantities or trajectories in space/time: so, one must specify the underlying \emph{physical geometry}. Third, the differential equations involved generally refer to \emph{coordinate representations} of these physical quantities with respect to some implicit coordinate chart, not to the original quantities.

These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult---at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $\R$-valued quantities $Q$ (that is, scalar fields), defined on $n$ (``spatial'' or ``temporal'') dimensions, taken to be isomorphic to the usual Euclidean space $\R^n$. For illustration, I give standard (in a sense, ``text-book'') formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions.

A theory, called trs, for Term Rewriting Systems in the theorem Prover PVS is described. This theory is built on the PVS libraries for finite sequences and sets and a previously developed PVS theory named ars for Abstract Reduction Systems which was built on the PVS libraries for sets. Theories for dealing with the structure of terms, for replacements and substitutions jointly with ars allow for adequate specifications of notions of term rewriting such as critical pairs and formalization of elaborated criteria from the theory of Term Rewriting Systems such as the Knuth-Bendix Critical Pair Theorem. On the other hand, ars specifies definitions and notions such as reduction, confluence and normal forms as well as non basic concepts such as Noetherianity.

The computation of sets of compatibles of incompletely specified finite-state machines (ISFSM's) is a key step in sequential synthesis. This paper presents implicit computations to obtain sets of maximal compatibles, compatibles, prime compatibles, implied sets, and class sets. The computations are implemented by means of BDD's that realize the characteristic functions of these sets. We have demonstrated with experiments from a variety of benchmarks that implicit techniques allow us to handle examples exhibiting a number of compatibles up to 2 1500 , an achievement outside the scope of programs based on explicit enumeration. We have shown, in practice, that ISFMS's with a very large number of compatibles may be produced as intermediate steps of logic synthesis algorithms, for instance, in the case of asynchronous synthesis. This shows that the proposed approach not only has a theoretical interest, but also practical relevance for current logic synthesis applications, as shown by its application to ISFSM state minimization.

The attitude to use whole numbers to point out notes in the twelve-tone chromatic scale dates back to about a century ago. Although the trend to quantify and to catalogue has always been present in Western music, it was with the dissolution of the tonal system that this trend became urgent, especially among those composers who were looking for new forms of expression.

RESUMEN La demostracion habitual del Segundo Teorema de Incompletud de Godel a partir de teorias debiles como IΣ1 es larga y tecnicamente intrincada. Raramente se dan todos los detalles y en muchos casos se omiten completamente apelando a la capacidad de lector para completarlos. En la primera parte de este articulo presentamos una guia de los principales puntos tecnicos de la demostracion habitual del Segundo Teorema de Incompletud de Godel a partir de teorias debiles. En la segunda parte presentamos una demostracion distinta y mas simple para la Teoria de Conjuntos de Zermelo-Fraenkel debida a T. Jech [Jech (1994)], y observamos que puede ser extendida de forma que englobe teorias debiles, evitando asi muchas de las complicaciones tecnicas que requieren las demostraciones habituales.

In the framework of the philosophy of contemporary mathematics, Hellman and Awodey both hold an interesting discussion on the role of Zermelo-Fraenkel Set Theory and Category Theory in the perspective of a good foundation for mathematics. For Hellman, neither Set Theory nor Category Theory constitutes a good foundational framework for mathematics and, in addition, Categories does not achieve a strong autonomy regarding Sets. Awodey’s claim is that Category Theory is a best option in the frame of a new way of understanding what a foundation of mathematics means. In this sense, the aim of this paper is to highlight the philosophical main features of this discussion, to establish some related positions and to show some interesting consequences for the philosophy of mathematics.

Multi-proxy approach was used to reconstruct the environmental conditions of remote lakes in the High Tatra Mountains (Slovakia) over the past few centuries (approximately 500-1000 years). Short sediment cores (*30 cm) taken from three morphologically similar glacial lakes distributed along altitudinal gradient (subalpine to alpine conditions) were analysed for organic matter content (LOI), diatoms and chironomids. Both descriptive and correlative approaches were used for analysing stratigraphical data. Predictive canonical correspondence analysis and co-correspondence analysis were applied to directly relate physical and biological proxies to each other. The relationship between LOI and biotic proxies was inconsistent across groups and lakes. Concordant patterns in diatom and chironomid composition were found in two non-acidified lakes. Common trends in those assemblages indicated major past environmental events such as the Little Ice Age, air pollution and lake acidification. In contrast, no relationship between the composition of diatom and chironomid assemblages was found in the formerly acidified lake, suggesting different responses of assemblages to acidification. While chironomids showed shifts that are attributable to recovery, diatoms assemblage remained relatively stable throughout the uppermost layers of the sediment record. On the other hand, climatic-driven changes in assemblages detected in the deeper layers were more pronounced in diatoms than in chironomids.

We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of that language that provides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these definitions, under various complexity measures.

We review different studies of the Periodic Law and the set of chemical elements from a mathematical point of view. This discussion covers the first attempts made in the 19 th century up to the present day. Mathematics employed to study the periodic system includes number theory, information theory, order theory, set theory and topology. Each theory used shows that it is possible to provide the Periodic Law with a mathematical structure. We also show that it is possible to study the chemical elements taking advantage of their phenomenological properties, and that it is not always necessary to reduce the concept of chemical elements to the quantum atomic concept to be able to find interpretations for the Periodic Law. Finally, a connection is noted between the lengths of the periods of the Periodic Law and the philosophical Pythagorean doctrine.

Although some nice properties of the Delaunay triangle-based structure have been exploited in many fingerprint authentication systems and satisfactory outcomes have been reported, most of these systems operate without template protection. In addition, the feature sets and similarity measures utilized in these systems are not suitable for existing template protection techniques. Moreover, local structural change caused by nonlinear distortion is often not considered adequately in these systems. In this paper, we propose a Delaunay quadranglebased fingerprint authentication system to deal with nonlinear distortion-induced local structural change that the Delaunay triangle-based structure suffers. Fixed-length and alignment-free feature vectors extracted from Delaunay quadrangles are less sensitive to nonlinear distortion and more discriminative than those from Delaunay triangles and can be applied to existing template protection directly. Furthermore, we propose to construct a unique topology code from each Delaunay quadrangle. Not only can this unique topology code help to carry out accurate local registration under distortion, but it also enhances the security of template data. Experimental results on public databases and security analysis show that the Delaunay quadrangle-based system with topology code can achieve better performance and higher security level than the Delaunay triangle-based system, the Delaunay quadrangle-based system without topology code, and some other similar systems.

We present a new strong normalisation proof for a λ-calculus with interleaving strictly positive inductive types λ µ which avoids the use of impredicative reasoning, i.e., the theorem of Knaster-Tarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based-a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone operators on the metalevel.

A comparison between five feature ranking methods based on entropy is presented on artificial and real datasets. Feature ranking method using ¡ £ ¢ statistics gives results that are very similar to the entropy-based methods. The quality of feature rankings obtained by these methods is evaluated using the decision tree and the nearest neighbor classifier with growing number of most important features. Significant differences are found in some cases, but there is no single best index that works best for all data and all classifiers. Therefore to be sure that a subset of features giving highest accuracy has been selected requires the use of many different indices.

In the book Defending the Axioms, Penelope Maddy proposes an approach to the philosophy of set theory based on her second philosophical method. In this essay, I evaluate some problems that this project runs into and propose some adjustments aimed at rescuing her view without making major concessions. 1

It is generally agreed that faces are not recognized only by utilizing some holistic search among all learned faces, but also through a feature analysis that aimed to specify more important features of each specific face. This paper addresses a novel decision strategy that efficiently uses both holistic and facial component (eye, nose and mouth) feature analysis to recognize faces. The proposed algorithm first performs a holistic search using the whole image features and selects probable candidates for further processing. Then the facial features of these probable candidates are compared and classified with the stored patterns in the train set. Finally the classification results are fused with a weighted majority voting to form the final decision. Simulation studies justify the superior performance of the proposed method as compared to that of Eigenface method.

A software model is a widely used technique to specify software. A UML model may contain different diagrams and a diagram is built from different elements. Each element is restraint to certain constraint or well-formedness rules (WFR). Assurance to these WFR is important to ensure the quality of UML diagrams produced. Even though, the formal definition to UML elements is rapidly increased; there is still lack of formalization of WFR. Therefore, this paper will define the WFR for use case diagrams as it is ranked as one of the most used diagram among UML practitioners. The formalization is based on set theory by logic and quantification. Based on an example of a use case diagram, we show how the diagram satisfied the WFR. Then, the elements involved in the well-formedness problem are detected and formally reasoned.

Program ćwiczeń z przedmiotu Logika I: I Rok Filozofii, rok akademicki 2016 / 2017 -grupa B 1) 21.11.2016. -KRZ -powtórka wiadomości: ujęcie semantyczne (różne metody testowania tautologiczności formuł KRZ) i syntaktyczne (dowodzenie tez KRZ, definiowanie funktorów prawdziwościowych). Literatura B. Stanosz Wprowadzenie do logiki formalnej. Podręcznik dla humanistów, Warszawa: PWN, 2010, Rozdział 2; Rachunek zdań, s. 25-57.

The emergence of big data analytics as a way of deriving insights from data brought excitement to mathematicians, statisticians, computer scientists and other professionals. However, the absence of a mathematical foundation for analytics has become a real challenge amidst the flock of big data marketing activities, especially in healthcare insurance. This paper developed a mathematical model for the analytics of healthcare insurance data using set theory. We implemented a prototype of the model using Java Programming Language, MapReduce Framework, Association Rule Mining and MongoDB. Also, we tested the model accuracy using data from the National Health Insurance Scheme in Nigeria to reduce delay. The result showed that the accuracy level was 97.14% on average, which depicts a higher performance for the model. This result implies that delays affecting the processing of data submitted by the providers and enrollees to the HMOs reduced drastically leading to the improvement in the eas...

Mathematics starts by introducing some simple concepts (in the founding cycle), which may seem self-sufficient. It is natural to start with a set theory that is not fully formalized as an axiomatic theory. It is usually presented as a popularized or implicit version of ZFC, admitting its axioms as necessary or obvious.

1981. A note on categoricity and completeness, History and Philosophy of Logic 2, 113–19.

Current study of axiomatic method presupposes concepts and results of string theory and set theory. But axiomatic method was vigorously pursued in the last half of the nineteenth century without explicit reference to the theory of strings, which was not axiomatized until the 1930s (independently by Tarski 1934 and by Hermes
1938). For historical remarks on string theory see Corcoran, Frank and Maloney1974. Moreover, although it is natural to read modern set-theoretic ideas in early methodological work, it is clear that early postulate theorists such as Dedekind, Peano, Hilbert, Huntington and Veblen were not working in any formal set-theoretic framework. Indeed, close correspondence between modern precise explications of methodological concepts and imprecise formulations found in the writings of early postulate theorists suggests that the creators of modern mathematical logic, particularly Carnap, Tarski, and Church, may have been guided by the goal of putting the early ideas on a more mathematically precise basis.

One of the most popular methods to visualize the overlap and differences between data sets is the Venn diagram. Venn diagrams are especially useful when they are ‘area-proportional’ i.e. the sizes of the circles and the overlaps correspond to the sizes of the data sets. In 2007, the BioVenn web interface was launched, which is being used by many researchers. However, this web implementation requires users to copy and paste (or upload) lists of IDs into the web browser, which is not always convenient and makes it difficult for researchers to create Venn diagrams ‘in batch’, or to automatically update the diagram when the source data changes. This is only possible by using software such as R or Python. This paper describes the BioVenn R and Python packages, which are very easy-to-use packages that can generate accurate area-proportional Venn diagrams of two or three circles directly from lists of (biological) IDs. The only required input is two or three lists of IDs. Optional paramete...

Many consumers consult online reviews before making (online) travel arrangements. Yet, little is known about the impact of these reviews on consumer decision making. This research applies consideration set theory to model the impact of online hotel reviews on consumer choice. An experimental study (N ¼ 168) that includes review valence (positive vs. negative reviews), hotel familiarity (well-known vs. lesserknown hotels), and reviewer expertise (expert vs. non-expert reviewers) as independent factors shows that on average, exposure to online reviews enhances hotel consideration in consumers. This is because positive as well as negative reviews increase consumer awareness of hotels, whereas positive reviews, in addition, improve attitudes toward hotels. These effects are stronger for lesser-known hotels. Reviewer expertise has only a minor -positive -influence on review impact.

Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al.

What shapes peace, and how can peace be successfully built in those countries affected by armed conflict? This paper examines peacebuilding in the aftermath of civil wars in order to identify the conditions for post-conflict peace. The field of civil war research is characterised by case studies, comparative analyses and quantitative research, which relate relatively little to each other. Furthermore, the complex dynamics of peacebuilding have hardly been investigated so far. Thus, the question remains of how best to enhance the prospects of a stable peace in post-conflict societies. Therefore, it is necessary to capture the dynamics of post-conflict peace. This paper aims at helping to narrow these research gaps by 1) presenting the benefits of set-theoretic methods for peace and conflict studies; 2) identifying remote conflict environment factors and proximate peacebuilding factors which have an influence on the peacebuilding process and 3) proposing a set-theoretic multi-method research approach in order to identify the causal structures and mechanisms underlying the complex realm of post-conflict peacebuilding. By implementing this transparent and systematic comparative approach, it will become possible to discover the dynamics of post-conflict peace.

In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of category theory, demonstrating their internal logic and veracity, their derivation and distinction from set theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds.

Previously unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of category theory. The book is vital to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers.

Lip reading has wide spread application, e.g. audiovisual Automatic Speech Recognition (A V-ASR), silent speech interface and person identification. Lip segmentation is one of important step in lip reading, because it provides basic information to be processed in subsequent steps. Lip tracking is a process of locating lip to associate lip in consecutive video fr ames. Chan-Vese model is a region-based segmentation algorithm, which also can be used as tracking method. This algorithm can detect boundary of object which not defined by gradient, while classical active contour can't be applied. This method also can detect object by any initial curve in the image, not necessarily surround the obj ect. This paper investigates about lip segmentation and tracking based on Chan-Vese model, preceded by the color segmentation.

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

Resumen. Los procesos cognitivos involucrados en la conceptualización, la categorización, y sus implicaciones para áreas como la lingüística y la lógica, son un tema recurrente en la filosofía contemporánea de la mente y del lenguaje, como en las ciencias del conocimiento. El presente artículo es un estudio exploratorio y preliminar sobre la categorización en el contexto del corpus aristotelicum, y a partir del libro inicial del Órganon: Categorías y sus referentes en otras obras como: Primeros y Segundos Analíticos, Tópicos, Del Alma, Física y Metafísica. El eje articulador es el análisis del triple estatus de las categorías aristotélicas: ontológico, lógico y lingüístico, que permite a su vez explorar el concepto sobre los universales y sus implicaciones para la lógica y la matemática contemporáneas.
Abstract. The cognitive processes involved in conceptualization, categorization, and their implications for areas such as linguistics and logic, are a recurring theme in the contemporary philosophy of mind and language, as in the sciences of knowledge. The present article is an exploratory and preliminary study on the categorization in the context of the corpus aristotelicum, and from the initial book of the Organon: "Categories" and their referents in other works as Analytica Priora and Posteriora, Topics, De Anima, Physics and Metaphysics. The articulating axis is the analysis of the triple status of the Aristotelian categories: ontological, logical and linguistic, which allows in turn to explore the concept of universals and their implications for contemporary logic and mathematics

Alex Ling seizes upon the philosophy of Alain Badiou to clarify a central question in film scholarship: ‘can cinema be thought?’ Treating this question on three levels, the author first asks if we can really think what cinema is, at an ontological level. Second, he investigates whether cinema can actually think for itself; that is, whether or not it is truly ‘artistic’. Finally he explores in what ways we can rethink the consequences of the fact that cinema thinks. In answering these questions, the author uses well-known films ranging from Hiroshima mon amour to Vertigo to The Matrix to illustrate Badiou’s philosophy as well as to consider the ways in which his work can be extended, critiqued and reframed with respect to the medium of cinema.